1 Answer
1

This is not a full solution, but it reduces the system to another recurrence relation which involves only coefficients. Let
$$f_n^{(k)}=a_0^k n(n-1)...(n-k+1)+a_1^k n(n-1)...(n-k)+...+a_k^k n(n-1)...(n-2k+1).$$
Then the above system reduces to
$$a_i^k={a_i^{k-1}\over i-k}$$
for $i < k$ and $a_k^k$ is determined by the condition that $f_{2k}^{(k)}=0$. Clearly the determination of $a_k^k$ is the difficult part. But if the sequence 3,19 etc. shows up
elsewhere, that should be suggestive.

Addendum: If all the $a_i^k$ are expressed in terms of $a_i^i$, then the recurrence relation for the
$a_k^k$ becomes the same as for the Taylor coefficients of the reciprocal Bessel function mentioned in Barry Cipra's link.